UY1: Magnetic Field Of A Circular Current Loop

This page gives the UY1 working model/result for Magnetic Field Of A Circular Current Loop. You reuse it when you build fields/potentials by symmetry or superposition, and when you connect fields to forces, energy, and circuits.

Module path: Electromagnetism (UY1)
Practice: UY1 Electromagnetism Quiz
Full routing: UY1 Assessment Map
Math toolkit: Mathematics for Undergraduate Physics

1) At a glance

Field at center of one circular loop:

Bcenter = μ₀ I/2R

For N turns: multiply by N.
On-axis field at distance z:

B(z) = μ₀ I R²/2(R² + z²)³/²

Modelling context: steady current (magnetostatics). The on-axis formula is for points on the symmetry axis; off-axis fields require a different calculation.

Prerequisites: Magnetic Field Of A Current Element
Next uses: Ampere’s Law, Force & Torque On Current Loop In Magnetic Field

2) Setup

Take a circular loop of radius R carrying steady current I.

Axis is line through center perpendicular to loop plane.
By symmetry, transverse field components cancel on axis.
Direction along axis from right-hand rule curled with current.

The axial formula B(z) is for points on the symmetry axis only.
Don’t forget the turn factor N for a coil.
Be consistent with the right-hand rule: reversing current reverses the field direction.

3) Core derivation/explanation

Biot-Savart integration for a loop gives axis field:

B(z) = μ₀ I R²/2(R² + z²)³/².

At z = 0:

Bcenter = μ₀ I/2R.

For N identical tightly wound turns:

BN(z) = N B(z).

Scaling insight:

Larger current increases field linearly.
Larger radius decreases center field (propto 1/R).

Checks (sanity)

At z = 0, the axis formula reduces to Bcenter = μ₀ I/(2R).
For zgg R, the field falls rapidly with distance (much faster than the 1/r field from a straight wire).

4) Worked example(s)

A 50-turn circular coil has R = 0.10 m and current I = 0.20 A. Find center field.

B = μ₀NI/2R = (4π × 10⁻⁷)(50)(0.20)/2(0.10) = 6.28 × 10⁻⁵ T.

5) Practice set (with hints + answers)

If I doubles, what happens to center field?
A single loop has R = 0.050 m, I = 3.0 A. Find Bcenter.
For fixed I,R, if turns increase from N to 3N, what happens to B?

Hints

Use direct proportionality with I and N.
Use Bcenter = μ₀I/(2R) for single-turn center.
Keep SI units in metres and amperes.

Answers

It doubles.
B = (4π × 10⁻⁷)(3.0)/(0.10) = 3.77 × 10⁻⁵ T.
It triples.

6) Summary + next steps

Circular loops produce concentrated axial fields.
Center and on-axis formulas are standard and widely used.
Turn count is a direct multiplier, which motivates coils/solenoids.

Next: Ampere’s Law Previous: Magnetic Field & Force Between Parallel Conductors Back To Electromagnetism